diff --git a/doc/whats_new/v0.20.rst b/doc/whats_new/v0.20.rst
index 0897f331ebda0..13734bb828660 100644
--- a/doc/whats_new/v0.20.rst
+++ b/doc/whats_new/v0.20.rst
@@ -181,3 +181,10 @@ Cluster
 - Deprecate ``pooling_func`` unused parameter in
   :class:`cluster.AgglomerativeClustering`. :issue:`9875` by :user:`Kumar Ashutosh
   <thechargedneutron>`.
+
+Changes to estimator checks
+---------------------------
+
+- Allow tests in :func:`estimator_checks.check_estimator` to test functions
+  that accept pairwise data.
+  :issue:`9701` by :user:`Kyle Johnson <gkjohns>`
diff --git a/sklearn/base.py b/sklearn/base.py
index 81c7e5dae7bcc..6f59cea3c7ab7 100644
--- a/sklearn/base.py
+++ b/sklearn/base.py
@@ -551,7 +551,6 @@ def is_classifier(estimator):
 def is_regressor(estimator):
     """Returns True if the given estimator is (probably) a regressor.
 
-
     Parameters
     ----------
     estimator : object
diff --git a/sklearn/neighbors/regression.py b/sklearn/neighbors/regression.py
index bd2ffb9b82489..b13f16cfd399e 100644
--- a/sklearn/neighbors/regression.py
+++ b/sklearn/neighbors/regression.py
@@ -9,6 +9,7 @@
 # License: BSD 3 clause (C) INRIA, University of Amsterdam
 
 import numpy as np
+from scipy.sparse import issparse
 
 from .base import _get_weights, _check_weights, NeighborsBase, KNeighborsMixin
 from .base import RadiusNeighborsMixin, SupervisedFloatMixin
@@ -139,6 +140,11 @@ def predict(self, X):
         y : array of int, shape = [n_samples] or [n_samples, n_outputs]
             Target values
         """
+        if issparse(X) and self.metric == 'precomputed':
+            raise ValueError(
+                "Sparse matrices not supported for prediction with "
+                "precomputed kernels. Densify your matrix."
+            )
         X = check_array(X, accept_sparse='csr')
 
         neigh_dist, neigh_ind = self.kneighbors(X)
diff --git a/sklearn/neighbors/tests/test_neighbors.py b/sklearn/neighbors/tests/test_neighbors.py
index 052c83c71d2e7..ceb53412018b8 100644
--- a/sklearn/neighbors/tests/test_neighbors.py
+++ b/sklearn/neighbors/tests/test_neighbors.py
@@ -2,7 +2,7 @@
 
 import numpy as np
 from scipy.sparse import (bsr_matrix, coo_matrix, csc_matrix, csr_matrix,
-                          dok_matrix, lil_matrix)
+                          dok_matrix, lil_matrix, issparse)
 
 from sklearn import metrics
 from sklearn import neighbors, datasets
@@ -731,10 +731,22 @@ def test_kneighbors_regressor_sparse(n_samples=40,
         knn = neighbors.KNeighborsRegressor(n_neighbors=n_neighbors,
                                             algorithm='auto')
         knn.fit(sparsemat(X), y)
+
+        knn_pre = neighbors.KNeighborsRegressor(n_neighbors=n_neighbors,
+                                                metric='precomputed')
+        knn_pre.fit(pairwise_distances(X, metric='euclidean'), y)
+
         for sparsev in SPARSE_OR_DENSE:
             X2 = sparsev(X)
             assert_true(np.mean(knn.predict(X2).round() == y) > 0.95)
 
+            X2_pre = sparsev(pairwise_distances(X, metric='euclidean'))
+            if issparse(sparsev(X2_pre)):
+                assert_raises(ValueError, knn_pre.predict, X2_pre)
+            else:
+                assert_true(
+                    np.mean(knn_pre.predict(X2_pre).round() == y) > 0.95)
+
 
 def test_neighbors_iris():
     # Sanity checks on the iris dataset
diff --git a/sklearn/utils/estimator_checks.py b/sklearn/utils/estimator_checks.py
index fdbecc358be35..40fcb1fdd069f 100644
--- a/sklearn/utils/estimator_checks.py
+++ b/sklearn/utils/estimator_checks.py
@@ -37,6 +37,7 @@
 
 from sklearn.base import (clone, TransformerMixin, ClusterMixin,
                           BaseEstimator, is_classifier, is_regressor)
+
 from sklearn.metrics import accuracy_score, adjusted_rand_score, f1_score
 
 from sklearn.random_projection import BaseRandomProjection
@@ -48,6 +49,8 @@
 from sklearn.exceptions import DataConversionWarning
 from sklearn.exceptions import SkipTestWarning
 from sklearn.model_selection import train_test_split
+from sklearn.metrics.pairwise import (rbf_kernel, linear_kernel,
+                                      pairwise_distances)
 
 from sklearn.utils import shuffle
 from sklearn.utils.fixes import signature
@@ -355,10 +358,56 @@ def _is_32bit():
     return struct.calcsize('P') * 8 == 32
 
 
+def _is_pairwise(estimator):
+    """Returns True if estimator has a _pairwise attribute set to True.
+
+    Parameters
+    ----------
+    estimator : object
+        Estimator object to test.
+
+    Returns
+    -------
+    out : bool
+        True if _pairwise is set to True and False otherwise.
+    """
+    return bool(getattr(estimator, "_pairwise", False))
+
+
+def _is_pairwise_metric(estimator):
+    """Returns True if estimator accepts pairwise metric.
+
+    Parameters
+    ----------
+    estimator : object
+        Estimator object to test.
+
+    Returns
+    -------
+    out : bool
+        True if _pairwise is set to True and False otherwise.
+    """
+    metric = getattr(estimator,  "metric", None)
+
+    return bool(metric == 'precomputed')
+
+
+def pairwise_estimator_convert_X(X, estimator, kernel=linear_kernel):
+
+    if _is_pairwise_metric(estimator):
+        return pairwise_distances(X, metric='euclidean')
+    if _is_pairwise(estimator):
+        return kernel(X, X)
+
+    return X
+
+
 def check_estimator_sparse_data(name, estimator_orig):
+
     rng = np.random.RandomState(0)
     X = rng.rand(40, 10)
     X[X < .8] = 0
+    X = pairwise_estimator_convert_X(X, estimator_orig)
     X_csr = sparse.csr_matrix(X)
     y = (4 * rng.rand(40)).astype(np.int)
     # catch deprecation warnings
@@ -383,8 +432,8 @@ def check_estimator_sparse_data(name, estimator_orig):
             if hasattr(estimator, 'predict_proba'):
                 probs = estimator.predict_proba(X)
                 assert_equal(probs.shape, (X.shape[0], 4))
-        except TypeError as e:
-            if 'sparse' not in repr(e):
+        except (TypeError, ValueError) as e:
+            if 'sparse' not in repr(e).lower():
                 print("Estimator %s doesn't seem to fail gracefully on "
                       "sparse data: error message state explicitly that "
                       "sparse input is not supported if this is not the case."
@@ -405,7 +454,8 @@ def check_sample_weights_pandas_series(name, estimator_orig):
     if has_fit_parameter(estimator, "sample_weight"):
         try:
             import pandas as pd
-            X = pd.DataFrame([[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3]])
+            X = np.array([[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3]])
+            X = pd.DataFrame(pairwise_estimator_convert_X(X, estimator_orig))
             y = pd.Series([1, 1, 1, 2, 2, 2])
             weights = pd.Series([1] * 6)
             try:
@@ -426,7 +476,8 @@ def check_sample_weights_list(name, estimator_orig):
     if has_fit_parameter(estimator_orig, "sample_weight"):
         estimator = clone(estimator_orig)
         rnd = np.random.RandomState(0)
-        X = rnd.uniform(size=(10, 3))
+        X = pairwise_estimator_convert_X(rnd.uniform(size=(10, 3)),
+                                         estimator_orig)
         y = np.arange(10) % 3
         y = multioutput_estimator_convert_y_2d(estimator, y)
         sample_weight = [3] * 10
@@ -438,7 +489,8 @@ def check_sample_weights_list(name, estimator_orig):
 def check_dtype_object(name, estimator_orig):
     # check that estimators treat dtype object as numeric if possible
     rng = np.random.RandomState(0)
-    X = rng.rand(40, 10).astype(object)
+    X = pairwise_estimator_convert_X(rng.rand(40, 10), estimator_orig)
+    X = X.astype(object)
     y = (X[:, 0] * 4).astype(np.int)
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
@@ -485,6 +537,8 @@ def check_dict_unchanged(name, estimator_orig):
     else:
         X = 2 * rnd.uniform(size=(20, 3))
 
+    X = pairwise_estimator_convert_X(X, estimator_orig)
+
     y = X[:, 0].astype(np.int)
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
@@ -522,6 +576,7 @@ def check_dont_overwrite_parameters(name, estimator_orig):
     estimator = clone(estimator_orig)
     rnd = np.random.RandomState(0)
     X = 3 * rnd.uniform(size=(20, 3))
+    X = pairwise_estimator_convert_X(X, estimator_orig)
     y = X[:, 0].astype(np.int)
     y = multioutput_estimator_convert_y_2d(estimator, y)
 
@@ -568,6 +623,7 @@ def check_fit2d_predict1d(name, estimator_orig):
     # check by fitting a 2d array and predicting with a 1d array
     rnd = np.random.RandomState(0)
     X = 3 * rnd.uniform(size=(20, 3))
+    X = pairwise_estimator_convert_X(X, estimator_orig)
     y = X[:, 0].astype(np.int)
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
@@ -621,6 +677,7 @@ def check_fit2d_1feature(name, estimator_orig):
     # informative message
     rnd = np.random.RandomState(0)
     X = 3 * rnd.uniform(size=(10, 1))
+    X = pairwise_estimator_convert_X(X, estimator_orig)
     y = X[:, 0].astype(np.int)
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
@@ -793,6 +850,7 @@ def check_pipeline_consistency(name, estimator_orig):
     X, y = make_blobs(n_samples=30, centers=[[0, 0, 0], [1, 1, 1]],
                       random_state=0, n_features=2, cluster_std=0.1)
     X -= X.min()
+    X = pairwise_estimator_convert_X(X, estimator_orig, kernel=rbf_kernel)
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
     set_random_state(estimator)
@@ -817,6 +875,7 @@ def check_fit_score_takes_y(name, estimator_orig):
     # in fit and score so they can be used in pipelines
     rnd = np.random.RandomState(0)
     X = rnd.uniform(size=(10, 3))
+    X = pairwise_estimator_convert_X(X, estimator_orig)
     y = np.arange(10) % 3
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
@@ -842,6 +901,7 @@ def check_fit_score_takes_y(name, estimator_orig):
 def check_estimators_dtypes(name, estimator_orig):
     rnd = np.random.RandomState(0)
     X_train_32 = 3 * rnd.uniform(size=(20, 5)).astype(np.float32)
+    X_train_32 = pairwise_estimator_convert_X(X_train_32, estimator_orig)
     X_train_64 = X_train_32.astype(np.float64)
     X_train_int_64 = X_train_32.astype(np.int64)
     X_train_int_32 = X_train_32.astype(np.int32)
@@ -887,7 +947,8 @@ def check_estimators_empty_data_messages(name, estimator_orig):
 def check_estimators_nan_inf(name, estimator_orig):
     # Checks that Estimator X's do not contain NaN or inf.
     rnd = np.random.RandomState(0)
-    X_train_finite = rnd.uniform(size=(10, 3))
+    X_train_finite = pairwise_estimator_convert_X(rnd.uniform(size=(10, 3)),
+                                                  estimator_orig)
     X_train_nan = rnd.uniform(size=(10, 3))
     X_train_nan[0, 0] = np.nan
     X_train_inf = rnd.uniform(size=(10, 3))
@@ -964,6 +1025,7 @@ def check_estimators_pickle(name, estimator_orig):
 
     # some estimators can't do features less than 0
     X -= X.min()
+    X = pairwise_estimator_convert_X(X, estimator_orig, kernel=rbf_kernel)
 
     estimator = clone(estimator_orig)
 
@@ -1138,6 +1200,7 @@ def check_classifiers_train(name, classifier_orig):
         classifier = clone(classifier_orig)
         if name in ['BernoulliNB', 'MultinomialNB', 'ComplementNB']:
             X -= X.min()
+        X = pairwise_estimator_convert_X(X, classifier_orig)
         set_random_state(classifier)
         # raises error on malformed input for fit
         with assert_raises(ValueError, msg="The classifer {} does not"
@@ -1159,11 +1222,18 @@ def check_classifiers_train(name, classifier_orig):
             assert_greater(accuracy_score(y, y_pred), 0.83)
 
         # raises error on malformed input for predict
-        with assert_raises(ValueError, msg="The classifier {} does not"
-                           " raise an error when the number of features "
-                           "in predict is different from the number of"
-                           " features in fit.".format(name)):
-            classifier.predict(X.T)
+        if _is_pairwise(classifier):
+            with assert_raises(ValueError, msg="The classifier {} does not"
+                               " raise an error when shape of X"
+                               "in predict is not equal to (n_test_samples,"
+                               "n_training_samples)".format(name)):
+                classifier.predict(X.reshape(-1, 1))
+        else:
+            with assert_raises(ValueError, msg="The classifier {} does not"
+                               " raise an error when the number of features "
+                               "in predict is different from the number of"
+                               " features in fit.".format(name)):
+                classifier.predict(X.T)
         if hasattr(classifier, "decision_function"):
             try:
                 # decision_function agrees with predict
@@ -1179,12 +1249,21 @@ def check_classifiers_train(name, classifier_orig):
                     assert_array_equal(np.argmax(decision, axis=1), y_pred)
 
                 # raises error on malformed input for decision_function
-                with assert_raises(ValueError, msg="The classifier {} does"
-                                   " not raise an error when the number of "
-                                   "features in decision_function is "
-                                   "different from the number of features"
-                                   " in fit.".format(name)):
-                    classifier.decision_function(X.T)
+                if _is_pairwise(classifier):
+                    with assert_raises(ValueError, msg="The classifier {} does"
+                                       " not raise an error when the  "
+                                       "shape of X in decision_function is "
+                                       "not equal to (n_test_samples, "
+                                       "n_training_samples) in fit."
+                                       .format(name)):
+                        classifier.decision_function(X.reshape(-1, 1))
+                else:
+                    with assert_raises(ValueError, msg="The classifier {} does"
+                                       " not raise an error when the number "
+                                       "of features in decision_function is "
+                                       "different from the number of features"
+                                       " in fit.".format(name)):
+                        classifier.decision_function(X.T)
             except NotImplementedError:
                 pass
         if hasattr(classifier, "predict_proba"):
@@ -1195,11 +1274,20 @@ def check_classifiers_train(name, classifier_orig):
             # check that probas for all classes sum to one
             assert_allclose(np.sum(y_prob, axis=1), np.ones(n_samples))
             # raises error on malformed input for predict_proba
-            with assert_raises(ValueError, msg="The classifier {} does not"
-                               " raise an error when the number of features "
-                               "in predict_proba is different from the number "
-                               "of features in fit.".format(name)):
-                classifier.predict_proba(X.T)
+            if _is_pairwise(classifier_orig):
+                with assert_raises(ValueError, msg="The classifier {} does not"
+                                   " raise an error when the shape of X"
+                                   "in predict_proba is not equal to "
+                                   "(n_test_samples, n_training_samples)."
+                                   .format(name)):
+                    classifier.predict_proba(X.reshape(-1, 1))
+            else:
+                with assert_raises(ValueError, msg="The classifier {} does not"
+                                   " raise an error when the number of "
+                                   "features in predict_proba is different "
+                                   "from the number of features in fit."
+                                   .format(name)):
+                    classifier.predict_proba(X.T)
             if hasattr(classifier, "predict_log_proba"):
                 # predict_log_proba is a transformation of predict_proba
                 y_log_prob = classifier.predict_log_proba(X)
@@ -1213,6 +1301,7 @@ def check_estimators_fit_returns_self(name, estimator_orig):
     X, y = make_blobs(random_state=0, n_samples=9, n_features=4)
     # some want non-negative input
     X -= X.min()
+    X = pairwise_estimator_convert_X(X, estimator_orig)
 
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
@@ -1260,7 +1349,7 @@ def check_supervised_y_2d(name, estimator_orig):
         # These only work on 2d, so this test makes no sense
         return
     rnd = np.random.RandomState(0)
-    X = rnd.uniform(size=(10, 3))
+    X = pairwise_estimator_convert_X(rnd.uniform(size=(10, 3)), estimator_orig)
     y = np.arange(10) % 3
     estimator = clone(estimator_orig)
     set_random_state(estimator)
@@ -1294,6 +1383,7 @@ def check_classifiers_classes(name, classifier_orig):
     # We need to make sure that we have non negative data, for things
     # like NMF
     X -= X.min() - .1
+    X = pairwise_estimator_convert_X(X, classifier_orig)
     y_names = np.array(["one", "two", "three"])[y]
 
     for y_names in [y_names, y_names.astype('O')]:
@@ -1325,7 +1415,7 @@ def check_classifiers_classes(name, classifier_orig):
 @ignore_warnings(category=(DeprecationWarning, FutureWarning))
 def check_regressors_int(name, regressor_orig):
     X, _ = _boston_subset()
-    X = X[:50]
+    X = pairwise_estimator_convert_X(X[:50], regressor_orig)
     rnd = np.random.RandomState(0)
     y = rnd.randint(3, size=X.shape[0])
     y = multioutput_estimator_convert_y_2d(regressor_orig, y)
@@ -1353,6 +1443,7 @@ def check_regressors_int(name, regressor_orig):
 @ignore_warnings(category=(DeprecationWarning, FutureWarning))
 def check_regressors_train(name, regressor_orig):
     X, y = _boston_subset()
+    X = pairwise_estimator_convert_X(X, regressor_orig)
     y = StandardScaler().fit_transform(y.reshape(-1, 1))  # X is already scaled
     y = y.ravel()
     regressor = clone(regressor_orig)
@@ -1429,6 +1520,12 @@ def check_class_weight_classifiers(name, classifier_orig):
         X, y = make_blobs(centers=n_centers, random_state=0, cluster_std=20)
         X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=.5,
                                                             random_state=0)
+
+        # can't use gram_if_pairwise() here, setting up gram matrix manually
+        if _is_pairwise(classifier_orig):
+            X_test = rbf_kernel(X_test, X_train)
+            X_train = rbf_kernel(X_train, X_train)
+
         n_centers = len(np.unique(y_train))
 
         if n_centers == 2:
@@ -1512,6 +1609,7 @@ def check_estimators_overwrite_params(name, estimator_orig):
     X, y = make_blobs(random_state=0, n_samples=9)
     # some want non-negative input
     X -= X.min()
+    X = pairwise_estimator_convert_X(X, estimator_orig, kernel=rbf_kernel)
     estimator = clone(estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator, y)
 
@@ -1586,6 +1684,7 @@ def check_sparsify_coefficients(name, estimator_orig):
 @ignore_warnings(category=DeprecationWarning)
 def check_classifier_data_not_an_array(name, estimator_orig):
     X = np.array([[3, 0], [0, 1], [0, 2], [1, 1], [1, 2], [2, 1]])
+    X = pairwise_estimator_convert_X(X, estimator_orig)
     y = [1, 1, 1, 2, 2, 2]
     y = multioutput_estimator_convert_y_2d(estimator_orig, y)
     check_estimators_data_not_an_array(name, estimator_orig, X, y)
@@ -1594,6 +1693,7 @@ def check_classifier_data_not_an_array(name, estimator_orig):
 @ignore_warnings(category=DeprecationWarning)
 def check_regressor_data_not_an_array(name, estimator_orig):
     X, y = _boston_subset(n_samples=50)
+    X = pairwise_estimator_convert_X(X, estimator_orig)
     y = multioutput_estimator_convert_y_2d(estimator_orig, y)
     check_estimators_data_not_an_array(name, estimator_orig, X, y)
 
diff --git a/sklearn/utils/tests/test_estimator_checks.py b/sklearn/utils/tests/test_estimator_checks.py
index 1b3a1ea7e597a..2323f8a634eb2 100644
--- a/sklearn/utils/tests/test_estimator_checks.py
+++ b/sklearn/utils/tests/test_estimator_checks.py
@@ -18,6 +18,8 @@
 from sklearn.cluster import MiniBatchKMeans
 from sklearn.decomposition import NMF
 from sklearn.linear_model import MultiTaskElasticNet
+from sklearn.svm import SVC
+from sklearn.neighbors import KNeighborsRegressor
 from sklearn.utils.validation import check_X_y, check_array
 
 
@@ -251,3 +253,16 @@ def __init__(self):
                         check_no_fit_attributes_set_in_init,
                         'estimator_name',
                         NonConformantEstimator)
+
+
+def test_check_estimator_pairwise():
+    # check that check_estimator() works on estimator with _pairwise
+    # kernel or  metric
+
+    # test precomputed kernel
+    est = SVC(kernel='precomputed')
+    check_estimator(est)
+
+    # test precomputed metric
+    est = KNeighborsRegressor(metric='precomputed')
+    check_estimator(est)